Problems by Edward Waring, M A. and Lucasian Professor of Mathematics in the University of Cambridge, F. R. S.

XLVI. Problems by Edward Waring, M A. and Lucasian Professor of Mathematics in the University of Cambridge, F. R. S. P R O. Read April 21, 1763. 1. Invenire, quot radices impossibiles habet data biquadratica æquatio \(x^4 + qx^2 - rx + s = 0\). 1mo Sit \(256s^3 - 128q^2s^2 + 144r^2q + 16q^4 \times s - 27r^4 - 4r^2q^3\) negativa quantitas, & duas & non plures impossibiles radices habet data æquatio. 2do Sit affirmativa quantitas, & vel —\(q\) vel \(q^2 - 4s\) negativa quantitas, & datæ æquationis quatuor radices erunt impossibiles. 3ti. Sit nihilo æqualis, & vel —\(q\) vel \(q^2 - 4s\) negativa quantitas, & datæ æquationis duæ inæquales radicis erunt impossibiles. 2. Invenire, quot radices impossibiles habet data æquatio \(x^5 + qx^3 - rx^2 + sx - t = 0\). 1mo Si signa terminorum æquationis \(w^{10} + 10qw^9 + 39q^2 + 10s \times w^8 + 80q^3 + 50qs + 25r^2 \times w^7 + 95q^4 + 124q^2s - 95s^2 + 92qr^2 + 200rt \times w^6 + 66q^5 - 360qs^2 + 196q^3s + 118qr + 260r^2s + 625t^2 + 400qrt \times w^5 + 25q^6 + 40s^3 - 53r^4 + 52q^3r^2 - 522q^2s^2 + 194q^4s + 708qr^2s + 240q^2rt + 1750q^3t - 950srt \times w^4 + 4q^7 + 106q^5s - 80q^3s^2 - 308q^3s^3 - 102qr^3 - 7q^2r^2 + 570r^2s^2 + 612q^2r^2s + 700r^3t - 3750t^2s + 2500t^2q^3 + 80rtq^3 - 2150qrst \times w^3 + 400s^3 - 360q^2s^3 - 15q^4s^2 + 24q^6s - 8q^5r^2\). \[ -45 q^2 r^4 - 270 r^4 s + 140 r^2 s q^3 + 960 r^2 s^2 q + 1875 t^2 r^2 + 1000 t r s^2 - 5000 t^2 q s + 1750 t^2 q^3 + 40 t r q^2 + 600 t r^3 q - 1650 t r s q^2 \times w^2 + 36 q^2 s^2 - 224 q^2 s^2 + 320 q s^4 + 4 q^3 r^4 + 27 r^6 - 40 r^2 s^2 + 434 r^2 q^2 s^2 - 24 r^2 s q^2 - 198 r^4 q s + 5000 t^2 s^2 - 450 t r^3 s - 6250 t^2 r + 675 r^4 - 3750 t^2 q^2 s + 3000 t^2 r^2 q + 600 t r^3 q^2 + 200 t r s^2 q - 330 t r q^3 s \times w + 3125 t^2 - 3750 q r t^2 + 2000 s^2 q + 2250 r^2 s - 900 s q^3 + 825 r^2 q^2 + 108 q^3 \times t^2 - 1600 s^3 r - 560 r q^2 s^2 - 16 r^3 q^3 + 630 r^3 q s + 72 r s q^4 - 108 r^5 \times t + 256 s^5 - 128 q^2 s^4 + 144 r^2 q s^3 + 16 q^3 s^3 - 27 r^4 s^2 - 4 r^2 q^3 s^2 = 0. \text{ continuo mutentur de } + \text{ in } —; & — in +; nullas impossibiles radices habet data æquatio. \] 2do. Si signa terminorum æquationis haud continuo mutentur de + in — & — in +; duæ vel quatuor datæ æquationis radices erunt impossibiles, prout ultimus ejus terminus fit negativa vel affirmativa quantitas. 3to. Si ultimus ejus terminus nihilo fit æqualis, & signa terminorum æquationis haud continuo mutentur de + in — &c — in +; tum vel quatuor vel duæ radices datæ æquationis erunt impossibiles, prout duo & non plures ultimi datæ æquationis termini nihilo sint æquales, necne. P R O. Sint \( x, y, v \), abscissa, ordinata &c area datæ curvæ, & fit \( y^n + a + bx \times y^{n-1} + c + dx + ex^2 \times y^{n-2} + f + gx + hx^2 + kx^3 \times y^{n-3} + \&c = 0 \). invenire, utrum area (\( v \)) quadrari potest, necne. Supponamus æquationem ad aream esse \( v^n + A + Bx + Cx^2 v^{n-1} + D + Ex + Fx^2 + Gx^3 + Hx^4 \times \) \[ v^{n-2} + I + Kx + Lx^2 + Mx^3 + Nx^4 + Ox^5 + Px^6 \] \[ \times v^{n-3} + \&c. = 0, \&c. \text{ consequenter erit } nyv^{n-1} \] \[ A + Bx + Cx^2yv^{n-2} + D + Ex + Fx^2 + Gx^3 + Hx^4 \] \[ \times yv^{n-3} + \&c. \] \[ \times v^{n-2} + \&c. \] Ex quibus æquationibus, si methodis notis exterminetur (v), habebimus æquationem, quæ exprimit relationem inter (x) & (y). Hujus autem æquationis coefficientes æquari debent coefficientibus datae æquationis \( y^n + ax^n + bx^{n-1} + cx^{n-2} + \&c. = 0; \&c. \) si quantitates A, B, C, \&c. exinde determinari possunt, curva quadratur, est enim \( v^n + A + Bx + Cx^2 \times v^{n-1} + D + Ex + Fx^2 + Gx^3 + Hx^4 \times v^{n-2} + \&c. = 0; \) aliter autem quadrari non potest. Ex. Sit data æquatio \( y^2 + x^2 - 1 = 0, \&c. \) supponamus æquationem ad aream \( v^2 + D + Ex + Fx^2 + Gx^3 + Hx^4 = 0; \&c. \) erit \( 2vy + E + 2Fx + 3Gx^2 + 4Hx^3 = 0, \) ita reducantur hæ duæ æquationes in unam, ut exterminatur (v), \&c. resultat æquatio \( y^2 + 16H^2x^6 + 24HGx^5 + 16HF + 9G^2x^4 + 8EH + 12FG \) \[ 4 \times Hx^4 + Gx^3 + Fx^2 + Ex + D \] \[ x^3 + 6GE + 4F^2x^2 + 4FEx + E^2 = 0; \] debet autem fractio \( \frac{16H^2x^6 + 24HGx^5 + 16HF + 9G^2x^4 + 8EH + 12FG}{4 \times Hx^4 + Gx^3 + Fx^2 + Ex + D} \) \[ x^3 + 6GE + 4F^2x^2 + 4FEx + E^2 \] esse \( x^2 - 1; \&c. \) consequenter \[4H = 16H^2\] \[4G = 24HG\] \[4F - 4H = 16HF + 9G^2\] \[4E - 4G = 8HE + 12FG\] \[4D - 4F = 6GE + 4F^2\] \[4E = 4FE\] \[4D = E^2\] sed e methodo communes divisores inveniendi constat has aequationes inter se contradictorias esse, & consequenter curvam haud generaliter esse quadrabilem. **THEO.** Sint \(x, y, v, abscissa \& ordinatae curvarum ABCD EFGHI \&c. \& A \beta y \delta \epsilon \&c. \& fit y = p x^n, \& v = \frac{n}{2 \cdot 3} p a^{n-1} x - \frac{n \times n-1 \times n-2}{30 \times 2 \times 3} p a^{n-3} x^3 + \frac{n \times n-1 \times n-2}{42 \times 2 \times 3} \times \frac{n-3 \times n-4}{4 \times 5} p a^{n-5} x^5 + \frac{n \times n-1 \times n-2 \times n-3 \times n-4 \times n-5}{30 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7} \times \frac{n-6}{66 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8} \times \frac{n-7}{691 \times n \times n-1 \times n-2 \times n-3} \times \frac{n-8}{2730 \times 2 \times 3 \times 4 \times 5 \times 6} \times \frac{n-9}{x^{n-9}} \times \frac{n-10}{x^{n-10}} \times \frac{n-11}{x^{n-11}} \times \cdots \&c. cujus ultimus terminus debet esse \(x^{n-1}\) vel \(x^{n-2}\), prout \((n)\) est par vel impar numerus. Sit \(x = AP = a\), biseetur \(AP\) in \(T\) in duas aequales partes, \& ducatur linea \(ET \delta\), \& si \(AE, EM, AM, jungantur; erit triangulum \(AEM = TP \epsilon \delta T\) areae. Deinde, Deinde, bisecentur TP, AT in R and V, & ducantur RG, CVγ; & jungantur AC, CE, EG, GM; & erunt duo triangula ACE + EGM = VTδγV areae. Eodem modo, si partes AV, VT, TR, RP iterum bisecentur in W, U, S, Q, &c ducantur lineae BWβ, UD, SF, QH; &c jungantur AB, BC, CD, DE, EF, FG, GH, HM; erunt quattuor triangula ABC + CDE + EFG + GHM = WVγβW areae; &c sic deinceps. Cor. 1. Si curva ABC & M sit conica parabola, \((c,e)y = p_i x^2\), erit \(v = \frac{1}{3} p a x\); & Aβγδ &c. erit recta linea; & propositio eadem est cum notissimâ propositione Archimedis de quadraturâ parabolæ. Cor. 2. Si \(y = p x^3\), erit \(v = \frac{1}{4} p a^2 x\), &c Aβγδ &c. iterum recta linea. Cor. 3. Datâ curvâ, cujus æquatio est \(y = p x^{2n}\), inveniri potest altera curva, cujus dimensiones sunt \((2n-1)\), in quâ summæ triangulorum ad singulas bisectiones erunt respectivè æquales summis triangulorum datæ curvæ. His adjici potest, quod si loco bisectionis abscissa AP aliâ quâvis ratione in æquales partes dividatur, summæ triangulorum curvæ ABCD &c. ad singulas divisiones æquales erunt segmentis curvæ Aβγδ &c.